Question

2) Suppose that you need to generate a random variable Y with a density function f (y) corresponding to a beta distribution with range [0,1], and with a non-integer shape parameter for the beta distribution. For this case there is no closed-form cdf or inverse cdf. Suppose your choices for generating Y are either: a) an acceptance-rejection strategy with a constant majorizing function g(u) = V over [0, 1], i.e., generate u1 and u2 IID from a U[0,1] generator and accept y = u1 if Vu2

Or b) use a numerical approximation to the inverse cdf of Y, say G, generate u from U[0,1] generator, and let y = G(u). Discuss the advantages and disadvantages of each approach. (10 points)

Answer 1

Advantages of acceptance-rejection method:

• As compared to inverse transform method, it requires neither CDF nor inverse of the function. And since in this particular question it is given that there is no closed dorm CDF or inverse CDF, it is appropriate to use this method as it will give a more efficient result.

Disadvantages of acceptance-rejection method:

• Finding a function g(x) with the same range and same shape to f(x) from which we need to generate random values can be difficult.
• It is a lengthy method to solve.

Advantages of acceptance-rejection method:

• It can be extended to the cases of truncated distribution. Following the same algorithm, instead of generating a random number u from a uniform distribution over the range[a,b], you can generate a uniform distribution which ranges from F(a) to F(b).

Disadvantages of Inverse transform method:

• Using this method requires us to estimate the inverse of a cumulative distribution function but since it is given in that there is no closed form CDF or inverse CDF, using this method would lead to an inefficient result.
• It is often a very slow method since it requires us a number of comparisons.